Can you help me with this problem 12/ Use the di vergencetheorem to eva lvate the fluxof the fleld F Cx,4,2)=through the Surface S, here S is the2.region x+y?KzK8=x²-y²2.HoF.ds=?S.

Name: Can you help me with this problem 12/ Use the di vergencetheorem to ev
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Description: Can you help me with this problem 12/ Use the di vergencetheorem to eva lvate the fluxof the fleld F Cx,4,2)=through the Surface S, here S is the2.region x+y?KzK8=x²-y²2.HoF.ds=?S.

Answered: 12/ Vse the divergence of the fleld…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Can you help me with this problem

12/ Use the di vergence
theorem to eva lvate the flux
of the fleld F Cx,4,2)=
through the Surface S, here S is the
2.
region x+y?KzK8=x²-y²
2.
HoF.ds=?
S.

Expert Solution

Step 1

To Determine: use the divergence theorem to evaluate the flux of the field

$\int \int_{s} F . d s = ?$

Given: we

$F (x, y, z) = < e^{z^{2}}, 9 y + \sin (x^{2} z), 9 z + \sqrt{x^{2} + 9 y^{2}} >$

Explanation: we have

through the surface S,where S is the region $x^{2} + y^{2} \leq z \leq 8 - x^{2} - y^{2}$ .

$d i v . F = \frac{\partial}{\partial x} (e^{z^{2}}) + \frac{\partial}{\partial y} (5 y + \sin (x^{2} z)) + \frac{\partial}{\partial z} (9 z + \sqrt{x^{2} + 9 y^{2}})$

$\Rightarrow d i v . F = 0 + 5 + 9 = 14$

using cylindrical coordinates as $x = r \cos θ, y = r \sin θ, z = z$ .

$0 \leq θ \leq 2 π, 0 \leq r \leq 2 and r^{2} \leq z \leq 8 - r^{2}$

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